A topology for the sets of shape morphisms

Cuchillo-Ibáñez, E.; Morón, Manuel A.; Portal, Francisco R. Ruiz del; Sanjurjo, José Manuel Rodríguez

doi:10.1016/s0166-8641(98)00024-8

Cited by 9 publications

(17 citation statements)

References 12 publications

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“…Also Cuchillo-Ibanez et al [6] constructed many generalized ultrametrics in the set of shape morphisms between topological spaces and obtained semivaluations and valuations on the groups of shape equivalences and kth shape groups. On the other hands, Cuchillo-Ibanez et al [7] introduced a topology on the set Sh(X, Y ), where X and Y are arbitrary topological spaces, in such a way that it extended topologically the construction given in [19]. Also, Moszynska [21] showed that the kth shape groupπ k (X, x), k ∈ N, is isomorphic to the set Sh((S k , * ), (Xx)) consists of all shape morphisms (S k , * ) → (X, x) with a group operation.…”

Section: Introductionmentioning

confidence: 98%

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On topological shape homotopy groups

Nasri

Ghanei

Mashayekhy

et al. 2016

Topology and its Applications

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In this paper, using the topology on the set of shape morphisms between arbitrary topological spaces X, Y , Sh(X, Y ), defined by Cuchillo-Ibanez et al. in 1999, we consider a topology on the shape homotopy groups of arbitrary topological spaces which make them Hausdorff topological groups. We then exhibit an example in whichπ top k succeeds in distinguishing the shape type of X and Y whileπ k fails, for all k ∈ N. Moreover, we present some basic properties of topological shape homotopy groups, among them commutativity ofπ top k with finite product of compact Hausdorff spaces. Finally, we consider a quotient topology on the kth shape group induced by the kth shape loop space and show that it coincides with the above topology.

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Section: Introductionmentioning

confidence: 98%

“…is an inverse limit of Sh((S k , * ), (X, x)) by [7,Theorem 2]. Since X λ 's are polyhedra,π k (X λ , x λ ) = π k (X λ , x λ ) which is finite for all λ ∈ Λ by the hypothesis.…”

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confidence: 99%

On topological shape homotopy groups

Nasri

Ghanei

Mashayekhy

et al. 2016

Topology and its Applications

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“…(b) In the last decade several papers dealing with (ultra)metric and topology structures on the (standard) shape morphism sets were written: [3,4,[15][16][17][18][19], . .…”

Section: Continuity Of the Hom-bifunctormentioning

confidence: 99%

Metrization of pro-morphism sets

Uglešić¹

2009

Glas. Mat.

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Abstract. Every pair of inverse systems X, Y in a category A, where Y is cofinite, admits a complete (ultra)metric structure on the set pro-A(X, Y ). The corresponding hom-bifunctor is not, generally, an internal Hom. However, there exists a subcategory of pro-A, containing tow-A, for which the hom-bifunctor is an invariant Hom into the category of complete metric spaces. Application to the sets tow-HcAN R(X, Y ) yields several new interesting results concerning Borsuk's quasi-equivalence.

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“…Cuchillo-Ibanez et al [9] introduced a topology on the set of shape morphisms between arbitrary topological spaces X, Y , Sh(X, Y ). Moszyńska [21] showed that for a compact Hausdorff space (X, x), the kth shape groupπ k (X, x), k ∈ N, is isomorphic to the set Sh((S k , * ), (X, x)) and Bilan [2] mentioned that the result can be extended for all topological spaces.…”

Section: Introductionmentioning

confidence: 99%

“…The aim of this paper is to introduce a topology on the coarse shape homotopy groupsπ * k (X, x) and to provide some topological properties of these groups. First, similar to the techniques in [9], we introduce a topology on the set of coarse shape morphisms Sh * (X, Y ), for arbitrary topological spaces X and Y . Several properties of this topology such continuity of the map Ω : Sh * (X, Y )×Sh * (Y, Z) −→ Sh * (X, Z) given by the composition Ω(F * , G * ) = G * • F * and the equality Sh * (X, Y ) = lim shape category [23,Theorem 2.2].…”

Section: Introductionmentioning

confidence: 99%

Topological coarse shape homotopy groups

Ghanei

Mirebrahimi

Mashayekhy

et al. 2017

Topology and its Applications

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Cuchillo-Ibanez et al. introduced a topology on the sets of shape morphisms between arbitrary topological spaces in 1999. In this paper, applying a similar idea, we introduce a topology on the set of coarse shape morphisms Sh * (X, Y ), for arbitrary topological spaces X and Y . In particular, we can consider a topology on the coarse shape homotopy group of a topological space (X, x), Sh, which makes it a Hausdorff topological group. Moreover, we study some properties of these topological coarse shape homotopoy groups such as second countability, movability and in particullar, we prove thatπ * top k preserves finite product of compact Hausdorff spaces. Also, we show that for a pointed topological space (X, x), π top k (X, x) can be embedded inπ * top k (X, x).

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A topology for the sets of shape morphisms

Cited by 9 publications

References 12 publications

On topological shape homotopy groups

On topological shape homotopy groups

Metrization of pro-morphism sets

Topological coarse shape homotopy groups

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